Capacitive bulk acoustic wave disk gyroscopes

ABSTRACT

Capacitive bulk acoustic wave x, y and z-axes gyroscopes implemented on (100) and (111) silicon substrates are disclosed. Exemplary gyroscopes comprise a handle substrate, a bulk acoustic wave resonator element supported by the handle substrate, and a plurality of electrodes surrounding and separated from the resonator element by very small capacitive gaps. The electrodes can excite and detect at least two degenerate bulk acoustic wave resonant modes in the resonator. Advantages include reduced size; higher Q, which improves noise and bias stability; larger bandwidth, and improved shock resistance. In addition, the high Q is maintained in atmospheric or near-atmospheric pressure which reduces the cost and complexity of the wafer-scale packaging of the gyroscope.

This application claims the benefit of U.S. Provisional Application No.60/786,304, filed Mar. 27, 2006.

FEDERALLY SPONSORED RESEARCH

This invention was made with Government support under Agreement No.ECS-0301900 awarded by the National Science Foundation of the UnitedStates. The Government has certain rights in this invention.

BACKGROUND

A gyroscope is a sensor that measures rate or angle of rotation.Micromachined gyroscopes constitute one of the fastest growing segmentsof the microsensor market. The application domain of these devices isquickly expanding from automotive to aerospace, consumer applications,and personal navigation systems. A multitude of applications exist inthe automotive sector including short-range navigation, anti-skid andsafety systems, roll-over detection, next generation airbag andanti-lock brake systems. Consumer electronics applications include imagestabilization in digital cameras, smart user interfaces in handhelds,gaming, and inertial pointing devices. Some applications requiresingle-axis gyroscope (Z-axis) and some require multiple axis rotationsensing (about X and Y and/or Z axes).

Miniature gyroscopes can be used for navigation. Inertial navigation isthe process of determining the position of a body in space by using themeasurements provided by accelerometers and gyroscopes installed on thebody. Inertial Measurement Units (IMU) for short-range navigation arevital components in aircraft, unmanned aerial vehicles, GPS augmentednavigation and personal heading references. An IMU typically uses threeaccelerometers and three gyroscopes placed along their respectiveorthogonal sensitive axes to gather information about an object'sdirection and heading. The components of acceleration and rotation ratecan consequently be interpreted to yield the object's accurate positionin space. An IMU is self-contained and can perform accurate short-termnavigation of a craft/object in the absence of global positioning system(GPS) assisted inertial navigation.

Current state-of-the-art micromachined vibrating gyroscopes operate atlow frequencies (ω₀=3-30 kHz) and rely on increased mass (M) andexcitation amplitude (q_(drive)) to reduce the noise floor and improvebias stability. If operated in 1-10 mTorr vacuum, such devices canachieve quality factors (Q) values on the order of 50,000 mainly limitedby thermoelastic damping in their flexures. It is known that thefundamental mechanical Brownian noise of a vibratory gyro is given by:

$\Omega_{z{({Brownian})}} \propto {\frac{1}{q_{drive}}\sqrt{\frac{4k_{B}T}{\omega_{0}{MQ}_{{Effect} - {Sense}}}}}$where q_(drive) is the drive amplitude; ω₀, M, and Q_(effect-sense) arethe natural frequency, mass and effective quality factor at the sensemode, respectively; k_(B) is the Boltzmann constant and T is theabsolute temperature.

Current state of the art micro-machined gyroscopes operate at arelatively low frequency (5-30 kHz) in their flexural modes and have a Qof less than 50,000 in high vacuum which results in a high noise floorwith limited mass. It would be desirable to reduce the noise floor ofvibrating gyros without having to increase the mass and drive amplitude,which is difficult to achieve in low power and small size. As will bedisclosed herein, a capacitive bulk acoustic wave gyroscope canaccomplish this task by (1) increasing the resonant frequency by 2 to 3orders of magnitude (to 2-8 MHz), and (2) increasing Q significantly byutilizing bulk acoustic modes that experience significantly lessthermoelastic damping compared to flexural modes. The very high Q of thebulk acoustic modes will translate into superior bias stability in thesegyros. Operation at high frequencies can increase the frequencybandwidth of the gyroscope by orders of magnitude, which decreases theresponse time of the sensors and relaxes the mode-matching requirements.Another benefit of increasing the resonant frequency of the gyro is inincreasing the stiffness of the device by orders of magnitude, whichtranslates into much higher shock resistance for the device (100 kGtolerance). In addition, the large stiffness of the device makes it lesssusceptible to air damping, which simplifies the packaging and reducesmanufacturing cost by eliminating the need for high vacuumencapsulation.

U.S. patents relating to gyroscopes include: U.S. Pat. No. 5,450,751issued to Putty, et al. entitled “Microstructure for vibratorygyroscope;” U.S. Pat. No. 6,128,954 issued to Jiang entitled “Spring fora resonance ring of an angular rate sensor;” U.S. Pat. No. 3,719,074issued to Lynch entitled “Hemispherical Resonator Gyroscope;” U.S. Pat.No. 4,793,195 issued to Koning entitled “Vibrating cylinder gyroscopeand method;” U.S. Pat. No. 6,848,304 issued to Geen entitled Six degreeof freedom micromachined microsensors;” and U.S. Pat. No. 6,837,108issued to Geen entitled “Micro-machined multi sensor providing 1-axis ofacceleration sensing and 2-axes of angular rate sensing.”

BRIEF DESCRIPTION OF THE DRAWINGS

The various features and advantages of the present invention may be morereadily understood with reference to the following detailed descriptiontaken in conjunction with the accompanying drawings, wherein likereference numerals designate like structural elements, and in which:

FIG. 1 illustrates an exemplary bulk acoustic wave gyroscope;

FIGS. 2 a and 2 b show ANSYS simulation data illustrating secondary andprimary elliptical modes of an exemplary bulk acoustic wave gyroscope;

FIG. 3 is a scanning electron microscope (SEM) view of a portion of anexemplary 800 μm diameter bulk acoustic wave gyroscope;

FIGS. 4 a and 4 b show frequency response of unmatched and matchedsecondary elliptical modes of an exemplary 800 μm diameter bulk acousticwave gyroscope;

FIG. 5 is a graph that shows measured sensitivity results of anexemplary reduced to practice 800 μm diameter bulk acoustic wavegyroscope in (100) silicon substrate;

FIG. 6 shows the frequency response of primary elliptical modes of anexemplary reduced to practice of 1200 μm diameter bulk acoustic wavegyroscope in (111) silicon substrate;

FIG. 7 is a graph that shows measured sensitivity results of anexemplary reduced to practice 1200 μm diameter bulk acoustic wavegyroscope in (111) silicon substrate;

FIG. 8 is a graph that shows a root Allan variance plot of primaryelliptical modes for an exemplary reduced to practice 1200 μm diameterbulk acoustic wave disk gyroscope in (111) silicon substrate;

FIGS. 9 a and 9 b show ANSYS simulation results for out of planedegenerative modes of an exemplary 800 μm diameter single crystalsilicon disk gyroscope; and

FIGS. 10-17 illustrate fabrication of an exemplary bulk acoustic wavegyroscope.

DETAILED DESCRIPTION

Referring to the drawing figures, disclosed herein are high frequency(MHz range) Z-axis and XY-axis Coriolis-based, capacitive bulk acousticwave gyroscope apparatus 10 or gyroscopes 10. As is illustrated in FIG.1, the gyroscopes 10 comprise a handle substrate 11, which may be asilicon-on-insulator (SOI) substrate 11. A resonator element 12(resonating disk 12 or resonating disk structure 12) is supported by aninsulating (buried oxide) layer 11 b of the handle substrate 11. Aplurality of electrodes 13 surround and are separated from the resonatorelement 12 by very small capacitive gaps 14. The electrodes 13 canexcite and detect at least two degenerate bulk acoustic wave resonantmodes in the resonator element 12. The resonator element 12 is generallya disk-like resonator element 12, which may be circular or polygonal.The resonator element 12 may be solid or perforated. The resonatorelement 12 does not have to be made out of a piezoelectric material. Infact, the preferred choice is a non-piezoelectric material, such assingle-crystalline or polycrystalline silicon. Other semiconducting,piezoelectric or metallic material such as silicon carbide, diamond,nano-crystalline diamond, gallium nitride, aluminum nitride, or quartzcan be used to make the resonator element. FIG. 3 is a scanning electronmicroscope (SEM) view of a portion of an exemplary 800 μm diametercapacitive bulk acoustic wave gyroscope 10 having a perforated resonatorelement 12. The plurality of electrodes 13 generally includes driveelectrodes 13 a, sense electrodes 13 b and electrostatic tuningelectrodes 13 c. The rest of electrodes 13 can be utilized to align thedegenerative bulk acoustic modes with the center of electrodes (i.e. tocancel the quadrature errors).

More particularly, exemplary 800 μm and 1200 μm diametercenter-supported single crystal silicon (SCS) perforated disk gyroscopes10 are disclosed. An exemplary 800 μm diameter disk gyroscope 10 wasimplemented on a 50 μm thick (100) single crystal silicon (SCS)substrate and was configured to be operated in high order ellipticmodes. The 1200 μm diameter disk gyroscope 10 was fabricated on a 35 μmthick (111) SCS substrate and was configured to be operated in primarilyelliptic modes. In both cases, (100) SCS and (111) SCS substrates arethe top layer (device layer) of SOI substrate 11. High aspect ratiotrenches 14 comprising the capacitive gaps 14 were realized using acombined polysilicon and single crystal-silicon micro-machining processknown in the art as HARPSS, implementing the capacitive disk gyroscopes10 on thick SOI substrates 11 (30-50 μm) with very small capacitive gaps14 (180-250 nm). Prototype bulk acoustic wave gyroscopes 10 show ultrahigh quality factor in excess of 100,000.

Exemplary bulk acoustic wave gyroscopes 10 may be implemented on asingle crystal silicon disk structure. The disk structure may have asolid or perforated configuration (FIG. 3). If a perforated disk 12 isused, symmetrical release holes 15 (shown in FIG. 3) are repeated every30° in (100) SCS substrate and every 45° in (111) SCS substrate tominimize the resonance frequency separation between the two degenerativemodes. The solid bulk acoustic wave disk gyroscope 10 is supported atits center with one or more suspended polysilicon traces 16 from thetop. The perforated bulk acoustic wave disk gyroscope can be supportedwith buried oxide 11 b of the SOI substrate 11 at the bottom. Also, thesuspended polysilicon trace 16 on the disk surface provides the DC biasto the disk 12. In order to capacitively excite and balance the highorder out-of-plane elliptical modes, twelve polysilicon electrodes 13,for example, extend over the top of the disk 12 at 30° intervals. Thesize of the capacitive gaps 14 between the extended polysiliconelectrodes 13 and the resonating disk 12 is the same as the verticalcapacitive gaps (typically 200 nm).

Two out-of-plane degenerative modes are available in SCS disk structuresat an identical resonance frequency. These two out-of-plane degenerativemodes are symmetric about the center of the disk 12 but 30° offcircumferentially in-plane. The top electrodes 13 are placed every 30°in-plane to detect and sense out-of-plane degenerative modes. When oneof the out-of-plane degenerative modes in FIG. 9 a is driven such thatits anti-node is aligned to the roll-axis (X-axis), upon application ofa roll rotation (rotation about X-axis) the energy will transfer fromthis first out-of-plane degenerative mode (FIG. 9 a) to the secondout-of-plane degenerative mode (FIG. 9 b). Consequently, the outputsignal due to the roll rotation can be measured at the electrodes 13that are located at the anti-nodes of the second degenerative mode (forexample, FIG. 9 b, D line). Since the first degenerative mode (FIG. 9 a,B-line) is at its zero displacement (node point) along the pitchrotation axis, there will be no transfer of energy from the firstout-of-plane degenerative mode (FIG. 9 a) to the second out-of-planedegenerative modes (FIG. 9 b) due to the pitch rotation (rotation aboutY-axis). As a result, if both pitch and roll rotations are appliedsimultaneously, the prototype technique can offer the solution toseparate the roll from pitch rotation. This procedure can be used tomeasure pitch rotation when the one of the out-of-plane degenerativemode (FIG. 9 b) is driven such that its anti-node is aligned to thepitch-axis (y-axis),and the other out-of-plane degenerative mode FIG. 9a is used to measure the output signal.

A version of the HARPSS process may be used to fabricate thecenter-supported SCS disk gyroscope 10 on 30 to 50 μm thick SOI wafers11. For gyroscopes 10 using bulk acoustic wave modes, the minimumdetectable rotation rate, which is normally limited by the electricalnoise, can be improved by orders of magnitudes over currently-availablevibratory microgyroscopes.

An advantage of the high frequency bulk acoustic wave gyroscope 10 is inreduction of the mechanical (Brownian) noise floor (3 to 4 orders ofmagnitude) due to an increase in the resonant frequency by 2 to 3 ordersof magnitude (to 2-10 MHz), and a significant increase in Q by utilizingbulk acoustic modes that experience less thermoelastic damping comparedto flexural modes. Further advantages of the high frequency bulkacoustic wave gyroscopes 10 are that they have: reduced size; higher Q,which improves noise performance and bias stability; larger bandwidth(BW=f/Q>25 Hz), and improved shock resistance. In addition, the high Qis maintained under atmospheric or near atmospheric pressure, whichsimplifies packaging of the gyroscopes 10 and reduces manufacturingcosts. The gyroscopes 10 can be operated at in-plane high orderdegenerative resonance modes which are different at the resonancefrequency from the out-of-plane degenerative modes. As a result, thegyroscopes 10 can be used to measure the yaw rotation as well as rolland pitch rotation at the different operating resonance frequencies.Finally, the design is not sensitive to variation in thickness of thebulk acoustic wave disk gyroscope 10 which in turn has advantages interm of manufacturability. A very unique feature in capacitive bulkacoustic wave disk gyroscopes 12 is that they are stationary devicescompared to conventional vibratory gyroscopes since the vibrationamplitudes is less than 20 nm due to their very small capacitive gaps(˜200 nm).

The capacitive bulk acoustic wave disk gyroscopes 10 operate in the MHzfrequency range, and are stationary devices with vibration amplitudesless than 20 nm, and achieve very high quality factors (Q) in moderatevacuum (and even atmospheric pressure), which substantially simplifiestheir wafer-level packaging. In addition, their much lower operating DCvoltages (Vp<5 V) and AC actuation voltages (160 mV) simplify theinterface circuit design and implementation using standard CMOSprocessing. Also, operating vibratory gyroscopes 10 at high frequenciesincreases the frequency bandwidth by orders of magnitudes compared tolow frequency mode-matched devices, which decreases the response time ofthe sensors and relaxes the mode-matching requirements.

As schematically shown in FIG. 1, the exemplary Coriolis-based bulkacoustic wave gyroscope 10 includes a center-supported disk structure 12(resonating element 12) with capacitively-coupled drive 13 a, sense 13 band control electrodes 13 c. The capacitive SCS bulk acoustic wave diskgyroscope 10 is designed to operate in either primary or secondarydegenerative elliptic modes.

FIGS. 2 a and 2 b show ANSYS simulations of elliptical modes of anexemplary bulk acoustic wave gyroscope 10. Due to the anisotropic natureof (100) single crystal silicon, only secondary elliptical modes of a(100) SCS disk that are spatially 30° apart have identical frequencies(FIG. 2 a). In (111) SCS disk gyroscopes 10, the primary ellipticalmodes of the disk resonator 12 (which are spatially 45° apart) haveidentical frequencies (FIG. 2 b). As a result, the electrodes 13 areplaced every 30° for (100) SCS or 45° for (111) SCS circumferentiallyaround the disk resonator 12 to maximize the sense and drivetransductions. In order to release the disk gyroscope 10 from the frontside, release holes are added to the disk structure. The release holes15 are repeated symmetrically every 30° in (100) silicon disk (or 45° in(111) silicon disk) to minimize any possible frequency split between thetwo degenerative elliptic modes.

One of the prominent design parameters in designing any vibratorygyroscope is the angular gain. The angular gain is defined as the ratioof the lag in the vibration pattern angle to the angle of rotation andit depends on the sensor structure as well as the resonant modeoperation. The angular gain was derived for solid disk structures and itis 1.8 times larger for primary elliptic modes (0.45) than the secondaryelliptic modes (0.24) in the disk gyroscopes 10. Although thesensitivity of (111) silicon disk gyroscope 10 is higher than thesimilar device in (100) silicon disk due to the larger angular gain,(100) silicon substrates 11 have advantages in terms of CMOScompatibility and supply availability compared to (111) single crystalsilicon.

Prototype gyroscopes 10 were fabricated on thick SOI wafers 11, orsubstrate 11 (30-50 μm-thick), using the HARPSS process. An exemplaryfabrication process flow is shown in FIGS. 10-17. In FIG. 10, a 2 μmthick sacrificial oxide mask 21 on an SOI substrate 11 (bottom layer 11a, insulating (buried oxide) layer 11 b, device layer 11 c) ispatterned. Deep trenches 22 are etched (FIG. 11) through the devicelayer 11 c to define the resonating SCS structures. In FIG. 12, thinlayer of sacrificial LPCVD oxide 23 is deposited that form capacitivegaps 14, and the trenches 22 are filled with LPCVD polysilicon 24subsequently. Next, the LPCVD polysilicon 24 is etched on the surfaceand the sacrificial oxide 23 is patterned on the surface (FIG. 13), anda LPCVD polysilicon layer 24 is deposited, doped and annealed (FIG. 14).After patterning (FIG. 15) the polysilicon on the surface to definepads, the polysilicon inside the trenches 22 and parts of the devicelayer 11 c are removed (FIG. 16) to define the electrodes 13. The deviceis then released in hydrogen fluoride (HF). The buried oxide layer 11 bof the SOI substrate 11 can be used to support the disk resonator 12 atthe bottom, which calls for careful timing of the HF release. Thepolysilicon trace 16 (FIG. 3) on the surface is used to provide a DCbias to the disk resonator 12. Also, each polysilicon electrode 13partially extends out on the disk structure 12 to provide anout-of-plane shock stop. In addition, the extended polysiliconelectrodes 13 can be used as in-plane electrodes in X-Y axis gyroscopes10 to excite and sense the out-of plane degenerative modes. As is shownin FIG. 17, PECVD oxide 27 is deposited and patterned and conductivematerial 28 (Aluminum, for example) is deposited to vacuum seal thegyroscope 10, in case very high performance is desired. This process iscompatible with Analog Device's SOIMEMS process discussed by M. W. Judy,“Evolution of Integrated Inertial MEMS Technology,” Solid-State Sensors,Actuators and Microsystems Workshop, Hilton Head Island, S.C., June2004, pp. 27-32, and may be integrated with CMOS electronics by addingsome pre- and post-CMOS fabrication steps.

Exemplary (100) silicon and (111) silicon disk gyroscopes 10 weretested. A sinusoidal drive signal was applied at the drive electrode 13a and output signal was monitored at sense electrode 13 b. The senseelectrode 13 b is located circumferentially off the drive electrode 13 aby 30° for (100) silicon and by 45° for (111) silicon disk gyroscope 10.Measurement results of an exemplary (100) silicon disk gyroscope 10 willnow be discussed. High-order elliptical modes of an exemplary 800 μmdiameter (100) disk gyroscope 10 were observed at 5.9 MHz with afrequency split of 300 Hz (FIG. 4). FIG. 4 shows the measured Q of125,000 and 100,000 of the high order elliptical modes for this devicein 1 mTorr vacuum. The corresponding Q values in 10 Torr vacuum werestill very high for this device (100,000 and 74,000).

A small initial frequency separation of 290 Hz between the drive andsense modes of this perforated device can be matched by the applicationof proper tuning voltages to tuning electrodes 13 c around the diskgyroscope 10. The matched-mode quality factor of the device was recordedto be 12,000. Mode-matching was achieved by applying a tuning voltage of10V DC. A large bandwidth (BW) of ˜490 Hz was measured for the bulkacoustic wave disk gyroscope 10 at frequency of 5.88 MHz which is 100times larger than low frequency mode-matched gyroscopes.

The output voltage from the exemplary gyroscope 10 was measured atdifferent angular speeds. The measured rate sensitivity of 800 μmdiameter (100) SCS disk gyroscope 10 is 0.19 mV/°/sec as is shown inFIG. 5, which is 17 times higher than that of the low frequencypolysilicon star gyroscope reported by M. F. Zaman, et. al., “TheResonating Star Gyroscope,” Proceedings IEEE Conference on MEMS, January2005, pp. 355-358.

Measurement results of a (111) silicon disk gyroscope 10 will now bediscussed. The primary elliptic modes of 1200 μm diameter disk gyroscope10 were observed less than 100 Hz apart without applying any tuningvoltages. The Q_(effective-sense) of (111) disk gyroscopes was 66,000and 58,000, in 1 mTorr and 1 Torr vacuum, respectively (FIG. 6).

The rate sensitivity response of an exemplary 1200 μm diameter (111) SCSdisk is presented in FIG. 7. The measured rate sensitivity of 1200 μmdiameter (111) bulk acoustic wave disk gyroscope 10 with discreteelectronics is 0.94 mV/°/sec which demonstrates higher rate sensitivitycompared to the (100) disk (0.20 mV/°/sec). This is expected due to thelarger angular gain and smaller frequency separation of the two ellipticmodes in the (111) disk 12.

Bias drift estimation will now be discussed. Gyro scale factor stabilityand bias drift are essential performance parameters in a gyroscope. Thescale factor stability is directly affected by the stability of theQ_(effect-sense) over time. It was observed that the measuredQ_(effect-sense) remained constant over a period of 24 hours at a fixedroom temperature and pressure. The zero rate output (ZRO) of the devicewas sampled. Using the collected ZRO data an Allan variance analysis wasperformed to characterize the long-term stability of the matched-modedevice interfaced with the discrete electronics. A root Allan varianceplot of an exemplary 1200 μm diameter (111) silicon disk gyroscope 10 isshown in FIG. 8. The measured bias instability of the gyroscope 10 is5.4°/hr (with less than 100 Hz mode separation). If desired, the tworesonance modes can be tuned and aligned by applying small DC voltages(<10V) to the tuning electrodes 13 around the disk, which translatesinto higher sensitivity and improved bias stability for devices.

Design specifications for an exemplary 1200 μm diameter vibratory bulkacoustic wave (111) silicon gyroscope 10 are summarized in Table 1. Inprototype designs, the minimum detectable rotation rate is limited bythe electronic noise which is mainly due to the high operatingfrequency. This problem can be solved by further increasing the gapaspect-ratio (AR>250) and use of very low noise amplifiers (V_(n)<100nV/√Hz).

TABLE 1 Summary of specifications for a 1200 μm diameter (111) SCS diskgyroscope. Device parameter Values Primary order elliptical modefrequency 2.90 MHz (ANSYS) 2.917 MHz (measured) Device thickness 35 μmCapacitive gap 180 nm DC polarization voltage 7 V Effective QualityFactor Qsense = 66,000 (measured) Theoretical mechanical resolution0.0442°/√hr Total noise 5.622°/hr/√Hz (measured) Bias instability5.4°/hr (measured) Sensor sensitivity 4.7 aF/°/s Rate sensitivity 0.94mV/°/s (measured)

Thus, bulk acoustic wave gyroscopes have been disclosed. It is to beunderstood that the above-described embodiments are merely illustrativeof some of the many specific embodiments that represent applications ofthe principles discussed above. Clearly, numerous and other arrangementscan be readily devised by those skilled in the art without departingfrom the scope of the invention.

1. Gyroscope apparatus, comprising: a substrate having a plane; a bulk acoustic resonator element; and a plurality of electrodes surrounding and separated from the resonator element by a very small capactive gaps, which electrodes can excite and detect at least two degenerate bulk acoustic wave resonant modes in the resonator; wherein the gyroscope apparatus senses rate or angle of rotation about at least one axis in the plane of the substrate.
 2. The apparatus recited in claim 1 wherein the substrate supports the bulk acoustic resonator element.
 3. The apparatus recited in claim 1 wherein the capacitive gaps are on the order of 200 nanometers or less.
 4. The apparatus recited in claim 1 further comprising direct current and alternating current voltage sources for the excitation and tuning of the apparatus.
 5. The apparatus recited in claim 1 wherein the resonator element is a disk-shaped structure made of a polysilicon or single-crystalline silicon.
 6. The apparatus recited in claim 1 wherein the resonator has a bulk acoustic wave resonant frequency which is at least 1 MHz.
 7. The apparatus recited in claim 1 further comprising support electronics for excitation, readout and turning of the resonator element.
 8. The apparatus recited in claim 1 wherein at least three bulk acoustic resonator elements and corresponding pluralities of electrodes are integrated on a single substrate to sense rate or angle of rotation about three orthogonal axes.
 9. Gyroscope apparatus, comprising: a handle substrate; a disk resonator element supported by the handle substrate; a plurality of electrodes surrounding and separated from the disk resonator element by very small capacitive gaps, which electrodes can excite and detect at least two degenerate bulk acoustic wave resonant modes in the disk resonator; and support electronics for excitation, readout and tuning of the disk resonator.
 10. The apparatus recited in claim 9 wherein the capacitive gaps are on the order of 200 nanometers or less.
 11. The apparatus recited in claim 9 wherein the disk resonator has a bulk acoustic wave resonant frequency which is at least 1 MHz.
 12. The apparatus recited in claim 9 wherein the disk resonator is supported by the handle substrate at its center.
 13. The apparatus recited in claim 9 wherein the disk resonator is aligned with a support element at a center of the disk resonator.
 14. The apparatus recited in claim 9 wherein the disk resonator is perforated.
 15. The apparatus recited in claim 9 which senses rate or angle of rotation about a vertical axis perpendicular to the plane of the substrate.
 16. The apparatus recited in claim 9 which senses rate or angle of rotation about at least one axis in the plane of the substrate.
 17. The apparatus recited in claim 9 wherein one or more resonator elements and one or more pluralities of electrodes and support electronics are integrated on a single substrate to form an integrated inertial measurement unit. 